In their paper, Durham–Fanoni–Vlamis make some observations about the cases of finite-invariance index 1, 2, and 3. For \(S) = 2\)  and \(S) = 3\) , they give examples showing that finite-invariance index is too coarse an invariant to determine whether \(S\)  has a good curve graph. On the other hand, they conjecture that \(S)=1\)  implies that \(S\)  has no good curve graph [1]}.
